Local derivation on some class of subspace lattice algebras

TL;DR

This paper proves that all local derivations on certain subspace lattice algebras constructed from reflexive lattices are actually derivations, extending understanding of algebraic structures in operator theory.

Contribution

It establishes that every local derivation on these specific subspace lattice algebras is a derivation, a new result in the structure theory of operator algebras.

Findings

Every local derivation from $Alg\mathcal{L}$ into $B(\mathscr{K})$ is a derivation.

Constructs specific classes of subspace lattices on direct sums of Hilbert spaces.

Extends the theory of derivations in the context of subspace lattice algebras.

Abstract

Let $\mathcal{H}$ be a separable Hilbert space and $\mathcal{L}_{0}\subset B(\mathcal{H})$ a complete reflexive lattice. Let $\mathscr{K}$ be the direct sum of $n_0$ copies of $\mathcal{H}$ ($n_{0}\in\mathbb{N}$ and $n_0\geq 2$) or the direct sum of countably infinite many copies of $\mathcal{H}$ respectively. We construct two class of subspace lattices $\mathcal{L}$ on $\mathscr{K}$. Let $Alg\mathcal{L}$ be the corresponding subspace lattice algebra. We show that every local derivation from $Alg\mathcal{L} $ into $B(\mathscr{K})$ is a derivation.